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A group action on cyclic compositions and $γ$-positivity

Shishuo Fu, Jie Yang

Abstract

Let $w_{n,k,m}$ be the number of Dyck paths of semilength $n$ with $k$ occurrences of $UD$ and $m$ occurrences of $UUD$. We establish in two ways a new interpretation of the numbers $w_{n,k,m}$ in terms of plane trees and internal nodes. The first way builds on a new characterization of plane trees that involves cyclic compositions. The second proof utilizes a known interpretation of $w_{n,k,m}$ in terms of plane trees and leaves, and a recent involution on plane trees constructed by Li, Lin, and Zhao. Moreover, a group action on the set of cyclic compositions (or equivalently, $2$-dominant compositions) is introduced, which amounts to give a combinatorial proof of the $γ$-positivity of the Narayana polynomial, as well as the $γ$-positivity of the polynomial $W_{2k+1,k}(t):=\sum_{1\le m\le k}w_{2k+1,k,m}t^m$ previously obtained by Bóna et al, with apparently new combinatorial interpretations of their $γ$-coefficients.

A group action on cyclic compositions and $γ$-positivity

Abstract

Let be the number of Dyck paths of semilength with occurrences of and occurrences of . We establish in two ways a new interpretation of the numbers in terms of plane trees and internal nodes. The first way builds on a new characterization of plane trees that involves cyclic compositions. The second proof utilizes a known interpretation of in terms of plane trees and leaves, and a recent involution on plane trees constructed by Li, Lin, and Zhao. Moreover, a group action on the set of cyclic compositions (or equivalently, -dominant compositions) is introduced, which amounts to give a combinatorial proof of the -positivity of the Narayana polynomial, as well as the -positivity of the polynomial previously obtained by Bóna et al, with apparently new combinatorial interpretations of their -coefficients.
Paper Structure (4 sections, 18 theorems, 37 equations, 3 figures, 1 table)

This paper contains 4 sections, 18 theorems, 37 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notations and Preliminary results
  3. Two proofs of Theorem \ref{['thm:tree-2']}
  4. A group action for cyclic compositions

Key Result

Theorem 1.1

The number of plane trees with $n$ edges, $k$ internal nodes, and $m$ internal nodes with degree larger than one is given by $w_{n,k,m}$.

Figures (3)

  • Figure 1: A Dyck path and its corresponding plane tree under the map $\theta$
  • Figure 2: A plane tree $T$ and all four of its preimages under $\phi$
  • Figure 3: A complete orbit under the action $\psi$

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 2.1: Bóna et al. BDL2022
  • Theorem 2.2: Bóna et al. BDL2022
  • Proposition 2.3
  • Lemma 2.4: Cycle lemma DM1947
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • ...and 28 more